Bandpass FIR filter properties can be obtained with conventional charge sampling circuits by controlling the integration/non-integration of a current proportional to an input signal. See, for example, “A 50-MHz CMOS Quadrature Charge Sampling Circuit With 66-dB SFDR,” S. Karvonen et al., IEEE International Symposium Circuits and Systems 2004, May 2004, Paper 11.5 (hereinafter, “non-patent document 1”). In a charge sampling circuit introduced in non-patent document 1, an input voltage is converted to current which is then stored in the form of electric charge by an integrator containing a capacitor and an amplifier. The storage period is controlled by a switch. Filtering becomes possible by alternatively turning on/off the switch according to a specified pattern. If a signal is sampled below the Nyquist frequency, aliased noise occurs in base band, degrading an SN ratio. In non-patent document 1, however, an input signal is sampled at a frequency beyond the Nyquist frequency and subjected to bandpass filtering before downsampling (decimation) to generate a low frequency output. In this instance, the frequency of a signal controlling the input switch is set to four times the frequency of the input signal. The output signal has a low frequency.
Non-patent document 1 introduces a scheme whereby a signal is subjected to Nyquist sampling after it is downconverted from a RF (carrier frequency) to a low frequency (IF). Before that circuit, a Gilbert mixer is needed for the RF-to-IF downconversion as in conventional art.
The scheme of non-patent document 1 attenuates the aliasing undesired signals caused by the downsampling by 18 dB. If one wants to apply the scheme to a tuner, a bandpass filter (BPF) with a sharp cutoff frequency profile needs be coupled to the input to filter out the undesired signals.
However, the charge sampling circuit of non-patent document 1 has a sampling frequency four times the carrier frequency. The circuit is difficult to make compliant with the latest communications standards which involve high carrier frequencies. There already exists a subsampling circuit which addresses these problems. The circuit implements subsampling below twice the carrier frequency. However, the sampling frequency should be higher than twice the maximum frequency of a base band signal based on which the carrier is modulated.
An example of the conventional charge subsampling mixer is shown in FIG. 21. See, for example, “A Discrete-Time Bluetooth Receiver in a 0.13 μm Digital CMOS Process,” K. Muhammad et al., 2004 IEEE International Solid-State Circuits Conference, February 2004, Paper 15.1 (hereinafter, “non-patent document 2”). A charge subsampling mixer 100 in FIG. 21 includes a transconductance stage 101, an input switch 102, two paths path_a and path_b, and an output capacitor 111. The transconductance stage (hereinafter, “gm stage”) 101 is a current source generating a current in proportion to the voltage value of a radio frequency (“RF”) input signal supplied at the input terminal IN. The paths path_a and path_b process signals.
The path path_a includes a switch 103, an integrating capacitor 107, a reset switch 105, and an output switch 109. The switch 103 opens/closes the path. The reset switch 105 discharges the integrating capacitor 107. The output switch 109 allows/disallows a voltage in proportion to the charge stored in the integrating capacitor 107 to be applied to an output terminal OUT.The path path_b includes a switch 104, an integrating capacitor 108, a reset switch 106, and an output switch 109. The switch 104 opens/closes the path. The reset switch 106 discharges the integrating capacitor 108. The output switch 109 allows/disallows a voltage in proportion to the charge stored in the integrating capacitor 108 to be applied to an output terminal OUT.
FIG. 22 shows signal waveforms, from a control circuit (not shown), controlling the switches in the charge subsampling mixer 100. When the signal level is 1 in the figure, the associated switch turns on; when the signal level is 0, the associated switch turns off. A signal LO, controlling the input switch 102, consists of pulses at 50% duty factor at the same frequency as the carrier frequency for the RF input signal. This frequency is designated a basic sampling frequency Fs for the system. An equivalent basic period to the frequency, termed Ts, is given by
  Ts  =      1    Fs  
Both a signal enable_a, controlling the switch 103 opening/closing the path path_a, and a signal enable_b, controlling the switch 104 opening/closing the path path_b, are a rectangular wave at a frequency of Fs/N where N is an integer greater than 1. The signals enable_a and enable_b are 1 for certain periods. To set the phase difference between the signals enable_a and enable_b to 180 degrees, the signal enable_b is off when the signal enable_a is on. When the signal enable_a (enable_b) for the path path_a (path_b) changes to 0, a signal out_a (out_b), controlling the output switch 109, changes to 1 and remains so for a period of N/2×Ts (=0.5×N/Fs). The signal reset_a controlling the reset switch 105, when the signal out_a (out_b) for the path path_a changes to 0, changes to 1 and remains so for a period of N/2×Ts (=0.5×N/Fs). The signal out_b controlling the output switch 110, when the signal enable_b for the path path_b changes to 0, changes to 1 and remains so for a period of N/2×Ts (=0.5×N/Fs). The signal reset_b controlling the reset switch 106, when the signal out_b for the path path_b changes to 0, changes to 1 and remains so for a period of N/2×Ts (=0.5×N/Fs).
The operational principles of the charge subsampling mixer 100 will be now described in reference to FIG. 23. When the signal enable_a is 1, if the signal LO changes to 1, the gm stage 101 supplies current to the integrating capacitor 107, changing a charge Qi stored in the integrating capacitor 107. The charge accumulated by the integrating capacitor 107 from time k×Ts to time (k+½)×Ts is given by
      Δ    ⁢                  ⁢          q      k        =                    ∫                  k          ·                      T            s                                                (                          k              +                              1                /                2                                      )                    ·                      T            s                              ⁢                        i          ⁡                      (            t            )                          ⁢                                  ⁢                  ⅆ          t                      =                  ∫                  -          ∞                          +          ∞                    ⁢                                    i            ⁡                          (              t              )                                ·                      γ            ⁡                          (                                                kT                  s                                -                t                            )                                      ⁢                                  ⁢                  ⅆ          t                    where i(t) is the output current from the gm stage 101, and γ(t) is the basic waveform of the signal LO in FIG. 24. The expression states the correlation between i(t) and γ(t), which is given byΔqk=[i{circle around (x)}Γ](kTs)Taking the Fourier transform of the expression, we obtainΔQk(f)=Ic(f)·Γ(f)·e−j2π·f·kTs=Ic(f)·Γ(f)·z−k where Ic(f) and Γ(f) are the Fourier transforms of i(t) and γ(t) respectively. In the expression, z is given byz=ej2π·f·Ts 
Γ(f) is given by
      Γ    ⁡          (      f      )        =                    ⅇ                  j          ⁢                                    T              s                        4                    ⁢          2          ⁢                      π            ·            f                              ⁢                                    ⅇ                          j              ⁢                                                T                  s                                4                            ⁢              2              ⁢                              π                ·                f                                              -                      ⅇ                                          -                j                            ⁢                                                T                  s                                4                            ⁢              2              ⁢                              π                ·                f                                                              j2π          ·          f                      =                  ⅇ                  j          ⁢                                    T              s                        2                    ⁢                      π            ·            f                              ⁢                        sin          ⁡                      (                                          π                ·                f                            ⁢                                                T                  s                                2                                      )                                    π          ·          f                    
The sinus cardinal (hereinafter, “sin c”) function is defined by
      sin    ⁢                  ⁢          c      ⁡              (        x        )              =            sin      ⁡              (        x        )              x  
Using the sin c function, Γ(f) is given by
      Γ    ⁡          (      f      )        =            ⅇ              j        ⁢                              T            s                    2                ⁢                  π          ·          f                      ⁢                  T        s            2        ⁢    sin    ⁢                  ⁢          c      ⁡              (                              π            ·            f                    ⁢                                    T              s                        2                          )            
From this expression, the Fourier transform of the charge accumulated from a time when LO changes to 1 to a time when it subsequently changes back to 0 is given by
      Δ    ⁢                  ⁢          Q      k        =            ⅇ              j        ⁢                              T            s                    2                ⁢                  π          ·          f                      ⁢                  T        s            2        ⁢    sin    ⁢                  ⁢                  c        ⁡                  (                                    π              ·              f                        ⁢                                          T                s                            2                                )                    ·              Ic        ⁡                  (          f          )                    ·              z                  -          k                    
While the signal enable_a is 1, the integrating capacitor 107 is charged N times. As the signal out_a changes to 1, the integrating capacitor 107 discharges completely. Furthermore, while the signal enable_b is 1, the integrating capacitor 108 is charged N times. As the signal out_b changes to 1, the integrating capacitor 108 discharges completely. In the waveform example in FIG. 22 (or FIG. 23), N=5. After the integrating capacitors 107, 108 output voltage, the reset signals reset_a and reset_b are changed to 1 to discharge the integrating capacitors 107, 108. The reset enables the capacitors 107, 108 to integrate from 0 every time. The charge stored in the capacitors 107, 108 is given byqout=Δq0+Δq1+Δq2+Δq3+Δq4 This is shown in FIG. 23. Using this expression, the Fourier transform of the output charge is given by
            Q      out        ⁡          (      f      )        =            ⅇ              j        ⁢                              T            s                    2                ⁢                  π          ·          f                      ⁢                  T        s            2        ⁢    sin    ⁢                  ⁢                  c        ⁢                  (                                    π              ·              f                        ⁢                                          T                s                            2                                )                    ·      Ic        ⁢                  (        f        )            ·              (                  1          +                      z                          -              1                                +                      z                          -              2                                +                      z                          -              3                                +                      z                          -              4                                      )            
It is understood that the expression represents a property (transfer function) of an FIR (finite impulse response) filter: FIR=(1+z−1+z−2+z−3+z−4). Hence, an FIR filter is realized by integrating the charge which forms the output current of the gm stage 1010, and unwanted signals are removed using that FIR filter.
In addition, letting Ci represent the capacitance of the integrating capacitor, the output voltage Vout(f) is given by
            V      out        ⁡          (      f      )        =                    Q        out            ⁡              (        f        )                    C      i      The relationship between the current Ic and the input voltage Vin(f) is given byIc(f)=gm·Vin(f)where gm is the transconductance of the gm stage 101.
In addition, transferring the charge stored in the integrating capacitors 107, 108 to the output capacitor 111 causes charge sharing. Therefore, the output voltage Vo(f) of the output capacitor 111 is given by
            V      o        ⁡          (      f      )        =                    z                  -          N                            1        +                                            C              o                                      C              i                                ⁢                      (                          1              -                              z                                  -                  N                                                      )                                ⁢                  V        out            ⁡              (        f        )            where Co is the capacitance of the output capacitor 111. This transfer function is a property of an IIR (infinite impulse response) filter.
From the expression, the frequency characteristic of the output voltage Vo (f) is given by
            V      o        ⁡          (      f      )        =            gm              C        i              ⁢                  T        s            2        ⁢    sin    ⁢                  ⁢                  c        ⁡                  (                                    π              ·              f                        ⁢                                          T                s                            2                                )                    ·      FIR      ·      IIR      ·                        V          in                ⁡                  (          f          )                    
The factor,
      ⅇ          j      ⁢                        T          s                2            ⁢              π        ·        f              ,does not affect the gain and is therefore neglected in the expression. The factor however affects the phase.
The zero point frequency for the FIR filter is equal to the aliasing frequency, lowering aliasing noise. Effects will be described in reference to FIGS. 25(a) to 25(e). The horizontal axes of all the graphs in FIGS. 25(a) to 25(e) show frequency. The vertical axes of all the graphs, except FIG. 25(c), show signal power. The vertical axis in FIG. 25(c) shows a FIR gain standardized with a maximum FIR gain. FIG. 25(a) is a signal spectrum at the input terminal of the gm stage 101. The signal spectrum demonstrates a desired signal, or a carrier frequency Fs, and noise. Integrating charge by the capacitors 107, 108 and sampling at a sampling frequency Fs generates a discrete time signal in the bandwidth 1/N times the sampling frequency Fs, as shown in FIG. 25(b). The desired signal aliases to DC, all noise is aliased to the bandwidth from 0 to Fs. FIG. 25(c) shows an FIR filter property realized by charge integration and resetting of the capacitors 107, 108. FIG. 25(d) is the signal spectrum in FIG. 25(b) after it is filtered by the FIR filter. Noise is filtered out at Fs/N, 2Fs/N, and other frequencies. FIG. 25(e) shows the signal spectrum in FIG. 25(d) after it is downsampled (the signal spectrum at the output terminal OUT of the charge sampling mixer 100). Noise is aliased to the bandwidth from 0 to Fs/N, but not in the signal bandwidth. The signal in FIG. 25(e) can be demodulated to obtain a base band signal.
The foregoing description is applicable in the absence of undesired signals and when the desired signal has a narrow bandwidth. If the desired signal has a wide bandwidth, like television broadcast waves (both analog and digital), noise is less attenuated on the edges of the bandwidth, with noise being aliased to the bandwidth of the desired signal. Furthermore, undesired signals are aliased, appearing at higher frequencies than the desired signal. Effects of these phenomena will be described in reference to FIGS. 26(a) to 26(e).
FIGS. 26(a) to 26(e) correspond to FIGS. 25(a) to 25(e). FIG. 26(a) assumes the presence of an undesired signal in addition to the desired signal. FIG. 26(b) shows the undesired signal being aliased the same way as the noise in FIGS. 25(a) to 25(e). FIG. 26(d) shows the aliased undesired signal being attenuated by the FIR filter. However, if the desired signal has a wide bandwidth, the undesired signal is insufficiently attenuated on the edges of the bandwidth of the aliased signal. The aliased undesired signal still has high power as shown in FIG. 26(e), making it difficult to retrieve the desired base band signal. To further reduce undesired signal aliasing, a filter with a sharp cutoff frequency profile is needed at the input of the subsampling mixer.
Raising the order of the FIR may possibly attenuate the undesired signal. However, to raise the order of the FIR, the downsampling factor N needs be increased, which in turn will narrow down the output bandwidth Fs/N. The bandwidth needs be twice that of the desired signal, placing limits on the increase of N. Should N be sufficiently large, the order of the IIR increases due to charge sharing, resulting in different gains in the bandwidth.